Ya. B. Zeldovich and Foundation of the Accretion Theory

Ya. B. Zeldovich and foundation of the accretion theory

Nikolay I. Shakura1, * 1Sternberg Astronomical Institute, Moscow M.V. Lomonosov State University, 13 Universitetskij pr., Moscow 119992, Russia; Kazan Federal University, Kremlevskaya 18, 420008 Kazan, Russia This short review is dedicated to academician Yakov Borisovich Zeldovich, the science of his epoch and the creation of modern accretion theory.

1. INTRODUCTION

Yakov Borisovich Zeldovich was born on March 8, 1914, in Minsk. His father Boris Naumovich Zeldovich was a lawyer, a member of a College of Lawyers. His mother Anna Petrovna Zeldovich (Kiveliovich) was a translator, a member of the Union of Writers. My first acquaintance with Ya. B. Zeldovich began with buying of his book “Further mathematics for beginners”. In the middle school my math teacher Alfred Viktorovich Baranovskiy explained us a method of finding extremum of parabola using Vi- Figure 2. Left panel is Zeldovich’s handbook “Further eta’s formulas. And he told us by the way that using mathematics for beginners”. Right panel is parabola of the methods of further mathematics allow to make extrema derivation using two methods: school method (that my teacher A. V. Baranovskiy taught me) and further math this process easier and more elegant. He did not tell method from Zeldovich’s book. Right upper corner is the any details and I was intrigued. And suddenly in a photo of Alfred Viktorovich Baranovskiy. book store of Bobruisk city (no far from my town) I saw that Ya. B. Zeldovich’s math book. I bought it hoping to find out the method of finding extrema of parabola. But few days later I went to pass en- trance exams to Faculty of Physics of the Moscow University. In 1963 I entered Astronomical Division of the Lomonosov Moscow State University (MSU). My de-

Figure 3. Professor B. A. Vorontsov-Velyaminov. arXiv:1809.11137v1 [physics.hist-ph] 28 Sep 2018

cision to study astronomy was inspired by another book that somehow found its way to Belarus of my childhood: “Essays About The Universe” written by Boris Aleksandrovich Vorontsov-Velyaminov, a pro- Figure 1. Parents of Ya. B. Zeldovich. fessor of astronomy at MSU. Later I attended his lectures as MSU student and was examined by him something I had never even dreamed about it two or three years ago when I studied his textbook “As- * Electronic address: [email protected] tronomy” in school. 2

Figure 5. Graphics portrait of Ya. B. Zeldovich by Figure 4. Ya. B. Zeldovich presents two values of his V. M. Lipunov, early 1980s. selected works to Pope John Paul II. At the center of the photo – professor Remo Ruﬃni, ICRANet director.

In my first three years at the MSU, Ya. B. Zel- dovich was nowhere in my surroundings, nor did I ever remember the book I had bought in Bobruisk. This book was not included in the list of recom- mended texts by no means because it was bad, but because the academician addressed it to beginning engineers and technicians as a self-education book on mathematics. Academician Ya. B. Zeldovich is known as an author of more than ten high-quality books on a wide range of topics in the sciences. For the first time I met Jakov Borisovich when the Dean’s office of Faculty of Physics organized a meeting of students with the editorial board of the Figure 6. Professor E. V. Shpolsky famous Soviet scientific journal Physics Uspeckhi. It was the third year of my studying in MSU. The Large Physics Auditorium being the venue. I was, O. Hahn and E. Strassmann [4], and theoretical pa- of course, very impressed with the brilliance of Ed- per by L. Meitner and O. R. Frisch [5]. These pa- uard Vladimirovich Shpolsky, the Editor-in-Chief. pers marked the begging of the new era of civilisa- Yakov Borisovich Zeldovich was silent all the time, tion development: the creation of powerful nuclear sitting with his densely hairy hands crossed under weapons. For the long time (1948–1965) Ya. B. Zel- his chin and being deep in his own minds. Later dovich was one of the leading member of the scien- on, already as his co-worker, I found out that he tific team of developing Soviet atomic project. When had never been a full-time undergraduate student he was asked about that time he drew a big black at any higher education institution. Having grad- square a “Black square” by Zeldovich. uated from a ten-year school in Leningrad in 1930, In the September of 1966, we, the fourth-year as- he started to work as a laboratory technician at the tronomy undergraduates of the Faculty of Physics, Institute of Mechanical Processing of Minerals for found ourselves enrolled in a new lecture course, some time and then at the Institute of Chemical “The structure and evolution of stars”, to be taught Physics (ICP) where, at the age of twenty (!), he be- by Ya. B. Zeldovich. Namely, at these lectures gan his postgraduate research under supervising of my first personal contact with the academician oc- Nikolay Nikolayevich Semyonov and where it took curred. The lectures were held on Fridays, and on him a fantastically short time to get to the very top Thursdays Ya. B. (some of other academicians called of the academic career ladder. Yakov Borisovich so) led a joint astrophysical sem- In late 1930-s Soviet Journal of Experimental inar (JAS) at the Sternberg Astronomical Institute and Theoretical Physics published several papers (SAI MSU), both for full-fledged scientists and for about uranium fission by Ya. B. Zeldovich and Yuliy young higher education graduates. Undergraduates Borisovich Khariton [1–3]. In the same time two did not have this seminar in their schedule and could papers about a division of uranium nuclei by free only drop in on when possible. When Ya. B. fin- neutrons were published: experimental paper by ished his first lecture, he asked if there were in the 3

Figure 7. Academician N. N. Semyonov, Nobel prizewinner.

Figure 9. Nobel prizewinner Otto Hahn and his colleagues: professors Fritz Strassmann, Lisa Meitner and Otto Frisch.

Figure 8. Academicians Y. B. Zeldovich and Y. B. Khariton, both of them were awarded three times as Heroes of Socialist Labour. audience students wishing to receive topics for their course theses and could they please stay on for a while. I was among those few who did. When my turn came, he asked me whether I had been at the JAS session the previous day and whether I had heard the talk on the (then mysterious) sources of cosmic X-ray radiation, and when I twice said yes, he made a suggestion. He oﬀered me to calculate the structure and spectrum of the radiation of a strong shock wave that arises near the surface of a neutron star due to gas falling onto the star.

2. BEGINNINGS OF X-RAY ASTRONOMY

In 1962, a group of American scientists led by Figure 10. Ya. B. Zeldovich with three golden medals of Prof. Riccardo Giacconi discovered first X-ray the Hero of Socialist Labour, that he got for Soviet atomic sources. Before that, astronomers had known the project. Malevich’s Black Square as a symbolic answer for questions about Zeldovich’s participation in the atomic only one X-ray source of extraterrestrial origin, project. namely, the solar corona. The coronal gas heated to million degrees due to some unknown mechanisms 4 produces X-ray emission. The luminosity of the solar corona in X-rays is approximately one millionth of the optical luminosity of the Sun (4 × 1033 erg/s). So it was natural to assume that other stars are sur- rounded by hot coronas too. However, simple calcu- lations showed that detectors available at that time could not reveal coronas even around the nearest stars on distances of several parsecs. Nevertheless, astronomers tried to detect X-ray radiation from the Moon! The Moon has no atmosphere, but likely some radiation can be produced as a luminescence of the Moon’s surface being at- tacked by the solar X-rays and charged particles. Thus, precisely at midnight of June 18, 1962, when Figure 11. Nobel prizewinner Antony Hewish and his the full Moon was shining, the “Aerobee” rocket was doctoral student Jocelyn Bell. launched. It reached a height of 225 km, its flight continued for 350 s and was quite successful: two of the three Geiger counters, with large surface and good sensitivity in the range 1.5−6 keV, were oper- ating during the flight time. In this energy range, the Earth’s atmosphere is totally opaque. Suddenly, instead of X-ray radiation from the Moon, a bright and before unknown source was discovered, which was far beyond the Solar system in the direction of the Scorpion constellation. It was named Sco X-1. In the following years, new rocket flights brought more and more discoveries of new X-ray sources. Gradually, a sky map covered with X-ray sources of different nature was created. First sources got their names according to directions of their location (Cyg X-1, Cyg X-2, Her X-1, Cen X-3, and so on). Later it was revealed that their X-ray luminosities were thousands or even tens of thousand times stronger Figure 12. Professor D. Ya. Martynov. than the solar luminosity in the optical range. The epoch of X-ray astronomy, the epoch of stunning discoveries in the Universe began. accreting neutron stars could explain observational According to own Ya. B. Zeldovich’s simple esti- data obtained with the launched equipment. mations, the shock wave originating when the gas This investigation was outlined in the article writ- surrounding a neutron star falls onto its surface ten by Ya. B. and me, the article was submitted should produce radiation primarily in the X-ray to “Soviet Astronomy” in 1968 and published in range. My goal was to carry out the full calcula- 1969 [6]. Meanwhile, in 1967 first radio pulsars were tion and investigate the process in detail. The main discovered which turned to be a strongly magnetized difficulty was connected with the following circum- rotating neutron stars [7]. This discovery starts new stance: the free-path length of a falling particle near epoch of radio astronomy: the epoch of study of var- the neutron star’s surface is much greater (tens of ious radio pulsars. But here I will discuss accreting times) than the characteristic timescale of interac- X-ray pulsars! tion between the radiation and matter. Usually it is As an undergraduate, I attended lectures on gen- not necessary to calculate the structure of the shock eral astrophysics by Dmitrii Yakovlevich Martynov, wave: it is sufficient to specify a change in density, then a Director of SAI, who paid special attention to pressure, temperature, and other physical parame- close binary stellar systems with the matter flowing ters depending on the velocity and the adiabatic in- from one star to another. Due to the relative orbital dex of the falling gas. In my problem, the density, motion of the two stars, this process results in an temperature and other parameters depended on en- origin of a disk-shape envelope around one of these ergy release in the braking zone. Moreover, plasma stars. It seemed natural to consider a binary stellar oscillations may arise in this zone. To describe them, system where one of the components is a neutron a consideration of kinetic plasma equations is re- star or even a black hole. quired instead of ordinary hydrodynamics. Finally, Thus, imagine a binary stellar system consisting it was shown that shock wave emission spectra from of a normal star and a black hole. The size of the 5

Figure 13. Two types of accretion disk formation in close binary systems with black holes or neutron stars.

Figure 14. Nobelprize winner Riccardo Giacconi and the map of X-ray sky obtained by Uhuru satellite [8]. ordinary star in this system is limited by that of the so-called Roche lobe. The normal star can increase in size as stellar evolution proceeds, and after the Roche lobe is filled, the matter starts flowing from stars in binary stellar systems was first detected in its surface to the region of gravitational attraction of the early 1970s by an experiment aboard the US the black hole (left panel of Fig. 13). Because of the purpose-built satellite Uhuru. The satellite was relative orbital motion of the binary components, launched into a circular orbit about 500 km high the matter does not fall directly onto the black hole from the Italian marine platform San Marco off the but forms a differentially rotating disk-like envelope coast of Kenya. The satellite got name Uhuru around it. Due to layer-to-layer friction, the matter (which is Swahili for freedom) due to the fact that that accumulates in the disk strongly heats up and the launch date, 12 December 1970, was Kenya’s In- starts glowing. In its rapid revolution, the matter dependence Day. Recognition is not always quick to in the disk slowly moves radially toward (or accretes come in science, and it was only in 2002 that the upon) the black hole, loosing its angular momentum mission project leader Riccardo Giacconi, a US as- in the process. The glowing of the disk is caused trophysicist of Italian origin, was awarded the Nobel by the matter accretion releasing its gravitational Prize in Physics for this pioneering work. A lot of energy. Indeed, the inner parts of the disk closest discovered by Uhuru X-ray sources are turned to to the black hole become hot enough to emit in the be accreting black holes and neutron stars. X-ray range. Virtually simultaneously with the discovery of ac- In a more complex accretion disk formation sce- creting black holes and neutrino stars in binary stel- nario, the optical companion does not fill its Roche lar systems, the foundation of the theory of disk ac- lobe and flows outward in all directions via stellar cretion on gravitating centers was laid by me under wind. In this case, a head-on shock is naturally ex- the guidance of Ya. B. The paper was submitted to pected to form in the region where the stellar wind “Soviet Astronomy” in June 1971 and published in stream is under the gravitational influence of the November 1972 [9]. Remarkable that the first pub- black hole. After the passage of the shock wave, lication with the results of Uhuru was received by the matter in the gravitational capture region of the editors of Astrophhysical Journal in 17th May 1971 black hole starts falling onto it – but not strictly and was published on July 1971 [10]. radially! Due to its rotational motion, the falling The bulk of the work was done together with matter has a specific angular momentum, which is Rashid Sunyaev. In another of our joint efforts, the somewhat greater than the specific orbital angular so-called standard model of disk accretion was de- momentum of the hole. In the case of the falling veloped and worked out in detail, which was pre- with conserved angular momentum the matter ac- sented [11] at the 55th Symposium of the Inter- quires the orbital motion at some distance from the national Astronomical Union held in May 1972 in black hole (right panel of Fig. 13). And then, well, Madrid. The symposium covered not only data from disk type accretion again! Uhuru but also the first theoretical results on mod- Replacing the black hole by a strongly magnetized eling the compact X-ray sources detected in binary neutron star in the binary system has a consequence stellar systems, i.e., accreting black holes and neu- that the stellar magnetic field starts destroying the tron stars. Our joint work was presented by Jim accretion disk at a distance of about a hundred Pringle (UK) – Rashid and I were then “nevyezdnye” star radii. The accreting matter then starts rapidly (not free to go abroad) – and was in fact an intro- falling along the magnetic force lines, encountering duction to a large scale paper [12] which was later the surface of the neutron star in the vicinity of the published in 1973 in the high-profile European jour- magnetic poles. Because magnetic and geographic nal Astronomy and Astrophysics, and on the basis poles are usually far apart, the rotation of the neu- of which Igor Dmitrievich Novikov and Kip Thorne tron star causes it to be observed as an accretion were able to calculate accurately the relativistic cor- pulsar. rections to radiation due to general relativity effects The presence of accreting black holes and neuron near black holes [13]. 6

Figure 17. Citation score of [12], NASA ADS.

Figure 15. Me and academician Rashid Alievich Sunyaev near the same blackboard in SAI, 70s and 2017.

Figure 18. Professor Donald Lynden-Bell.

i.e., forty five years on, exceeding 8500. The study of accretion disks has led to the discovery in the cores of active galaxies and in quasars of supermassive black holes with masses ranging from a few dozen to a few hundred millions solar masses. The first theoretical results on the glowing of accretion disks around supermassive black holes were published by David Lynden-Bell (UK) in 1969 in Nature [14]. Figure 16. Nobelprize winner K. Thorne and In the last 45 years a great advance has been made corresponding member of the Russian Academy of Sciences in observations, as well as in the theory of accretion I. D. Novikov on the Prof. Leonid Petrovich Grishchuk disks. Now accretion disks is a specific extensive memorial conference, November 2016. branch of astrophysics, important to the same degree as the structure and evolution of stars, interstellar medium, etc. The basis for the modern theory of ac- The results Uhuru produced during its three cretion disks is: a) cosmic (magneto)hydrodynamics, years of operation were spectacular. Not only a large b) radiative transfer, c) general relativity effects in number (339) of newly discovered X-ray sources were the vicinity of compact stars. catalogued, but Uhuru also provided guidance for But what about Moon? “Did R. Giacconi and other space observatories. Currently, space X-ray his colleagues miss their target?”, you can ask me. sources of various natures (not necessarily accreting Lunar X-ray light was discovered later by satellite relativistic stars in binary stellar systems!) number ROSAT, that contains more perfect X-ray telescope in the hundreds of thousands. with grazing incidence mirrors. Rashid’s and my pioneering paper [12] gained wide More detailed historical review was published in popularity; its number of citations as of July 2018, the year of Ya. B.’s 100 anniversary in “Physics Us- 7

Figure 20. z-component of gravitational acceleration.

main terms. For geometrically thin accretion disks Keplerian rotation law takes place: r v GM ω = ϕ = . (3) r r3

Figure 19. Professor Joachim Trumper, the PI of ROSAT The second Euler’s equation (the equation of an- and my close friend in Germany. gular momentum):

∂(ω r2) 1 ∂ Σ v = (W r2) , (4) pekhi” [15]. 0 r ∂r r ∂r rϕ

R z0 where Wrϕ = wrϕ dz is height-integrated vis- −z0 3. STANDARD MODEL OF DISK ACCRETION cous shear stress. The equation of hydrostatic equilibrium along z- coordinate (see Fig. 20): Many excellent books and proceedings on accretion disk theory have been published so far. I espe- 1 ∂P GM = − z . (5) cially want to mention two of them. Namely, “Ac- ρ ∂z r3 cretion Power in Astrophysics” by Juhan Frank, An- drew King and Derek Raine [16] and “Black Hole Ac- cretion Disks. Towards a New Paradigm” by Shoji Kato, Jun Fukue and Sin Mineshige [17]. Bellow the 3.2. The equation of the energy balance theory of axisymmetric geometry thin and optically thick accretion disks (a standard Shakura–Sunyaev α-model) will be briefly outlined. ∂S Σ T v = Q+ − Q− , (6) 0 r ∂r 3.1. Main equations where T is the typical (along z-coordinate) accreting matter temperature, S is the entropy per unit + − The height-integrated continuity equation for the mass, Q and Q are the vertical-integrated rate of disk layer on some radius r: thermal energy release by viscous shear and rate of thermal energy losses by radiation. Left-hand side ∂Σ 1 ∂ of the equation describes advection energy transfer, 0 = − (Σ v r) , (1) ∂t r ∂r 0 r for geometrically thin disks this value is small and Q+ = Q−. where vr is the radial velocity of the accreting mat- R z0 ter, Σ0 = ρ dz is surface density of accretion −z0 disk, ρ is the accreting matter density, z0 is the semi- 3.3. Viscosity height of the disk and z and t are coordinate along disk axis and time correspondingly. The main challenge of solving the disk accretion The first Euler’s equation: problem is to account for viscous forces. Usually, viscosity of ionized plasma is very low. That is why ∂v GM 1 ∂P r 2 it is necessary to assume the presence of developed vr − ω r = − 2 − + ... (2) ∂r r ρ ∂r turbulence or magnetic fields in the accretion disk. As the first step, let us introduce turbulence viscos- Here ω is angular velocity of accreting mater, M ity: is the mass of central object (i.e., black hole), P is the accreting matter pressure and ellipsis denotes ∂ω ∂ω wt = ν ρ r = ρ < v l > r , (7) other terms, such as viscosity, small compared to rϕ t ∂r t t ∂r 8 where νt is turbulent kinematic viscosity, vt and lt are turbulent speed and length. Using empirical Prandtl’s law vt = lt r|∂ω/∂r| for Keplerian disk we can rewrite:

t 2 2 wrϕ = −ρ vt = −α ρ vs , (8) where vs is the sound velocity of the accreting matter. Here we introduce the α-parameter:

v 2 α = (Mt)2 = t , (9) vs where Mt is the turbulent Mach number.

Figure 21. Spectral distribution of radiative flux density 3.4. Energy generation and transfer from a standard optically thick, geometrically thin disk in the Newtonian metric. The horizontal axis shows the normalised radiation frequency. The vertical axis shows The energy release occurs due to the differential normalized spectral radiative flux density in units of 2 rotation of the viscous turbulent disk, the energy [erg/Hz/cm /s]. Here h is the Planck constant, kB is the 2 release rate per unit volume is Boltzmann constant, Tmax = Teff r = (7/6) Rin is the maximum surface temperature of the standard disk. ∂ω 3 = r w = − ω w . (10) rϕ ∂r 2 rϕ equations [12]. Using the vertical structure solution The vertical gradient of the radiation flux q is pro- analytical relations for radial distribution of various portional to the energy release rate: parameters of accretion disk can be obtained, see, ∂q i.e., [20, 21]. = . (11) ∂z In the most simple case when radiation from the disk surface is black-body the radiation flux is The equation transfer equation in the diffusion approximation: r ! 1 ∂ω 3 GM Rin Q = r Wrϕ = M˙ 1 − c ∂(a T 4) 2 ∂r 8π r3 r (14) r = −q , (12) 3 ρ ∂z 4 κ = σSB Teff , where κ is the opacity coefficient and ar is the radiation constant. where σSB is the Stefan–Boltzmann constant. The spectral radiative flux in a unit solid angle from a flat accretion disk at distance d from the disk 3.5. Vertical and radial disk structure is equal to 2 π Z Rout Equations (5), (10) and (11) with the continuity F = cos i I r dr , (15) ν d2 ν equation in the form of ∂σ/∂z = ρ form the sys- Rin tem of ordinary differential equations, its solution where ν is radiation frequency, i is the inclination of for right boundary conditions provides vertical disk the disk to the line of sight, R is the outer disk structure. This problem is like the problem of cal- out radius and I (r) is the intensity of radiation from culation of stellar structure, and can be solved nu- ν the disk surface. merically, see, i.e., [18, 19]. With the assumption of black-body radiation In the case of stationary accretion disk, when ac- Eq. (15) gives spectral disk flux shown in Fig. 21. cretion rate M˙ = 2π Σ v r is constant, we can in- 0 r If the case of electron scattering spectrum has more tegrate Eq. (4) and obtain radial dependence of vis- complicated shape, see [22] for details. cous shear: r ! M˙ ω R W = 1 − in , (13) rϕ 2π r 3.6. Non-stationary disk accretion where Rin is the inner disk radius, that equals in- Substituting Eq. (4) to Eq. (1) gives us equation of nermost stable circular orbit for the case of accre- viscous disk evolution. It is convenient to introduce tion on the black hole. Thus, radial structure of ac- a variable F = −Wrϕ, henceforth 2 π F means the cretion disk can be found as a solution of algebraic total momentum of viscous forces acting between the 9 adjacent layers, and an independent variable h = ω r2 (Keplerian angular momentum per unit mass). Then we have: M˙ (r, t) ∂F = − (16) 2 π ∂h and ∂Σ 1 (GM)2 ∂2F 0 = . (17) ∂t 2 h3 ∂h2 Figure 22. X-ray light curve of 1975 outburst of A0620−00 [30]. Dots are observations, line is numerical A relation between Σ0 and F depends on the vertical model of accretion disk evolution. structure, i.e. on opacity law, value of α, etc. This equation is a non-linear partial differential equation of diffusion that in the general case can be solved only numerically. For the case of power-law opacity κ(ρ, T ) = a b κ0 ρ /T : (GM)2F 1−m Σ = , (18) 0 2(1 − m) D h3−n Figure 23. Temporal evolution of accretion rate on the where m and n are dimensionless coefficients that de- black hole in 2002 outburst of 4U1543−47 [31]. pend on a and b, and D is dimension coefficient [20]. For this relation between Σ0 and F the equation of viscous evolution (16) has the following form The modern astrophysics is largely impacted by ∂F F m ∂2F Zeldovich’s papers on the interaction of matter and = D . (19) ∂t hn ∂h2 radiation in the Universe and on formation of large- scale structure of the Universe. Yakov Borisovich There are a number of self-similar solution of this Zeldovich died on the 2nd of December 1987. equation, see [20, 23–25]. For the case of m = 0 I thank Konstantin Malanchev and Natalia Green functions of this problem were found in [26, Shakura for preparation of the manuscript. 27]. The theory of non-stationary accretion applies successfully to X-ray outbursts of accretion disks in close binary systems with black holes or neutron stars. Often such outbursts cannot be described in the terms of self-similar solutions of the viscous equation (16). On the other hand numerical solution of this equation provides possibility to take into account effects of disk self-irradiation and hydrogen re- combination. The examples of such numerical mod- eling are shown in Fig. 22 and 23. These simulations showed that α-parameter is not small, it is about 0.7. In the early 80-s Ya. B. analysed the results of Sir Taylor’s laboratory experiment [28] and explained generation of turbulence in it [29]. Several words about this experiment. Two coaxial cylinders were taken, the gap between them was filled with some liquid. The bigger cylinder was rotated with acceleration, the smaller one was stable. At some moment the liquid became turbulent. It was found that the bigger gap between cylinders required larger rotation velocity to arise turbulence. This relation was connected with critical Reynold’s number. Ya. B. Zel- dovich suggested the theoretical basement which explained these empirical data obtained in Taylor’s ex- Figure 24. The critical Reynolds number dependence on periment. Nevertheless, Ya. B. pointed out that this the gap’s width t/R1. Taylor’s experimental results and the result cannot be applied to the disk accretion. exponential approximation by Ya. B. are shown [29]. 10

4. INSTEAD OF EPILOGUE

From the resolution of the Presidium of the Rus- sian Academy of Sciences of 11 Feb. 2014

• Establish the Academician Ya. B. Zel- dovich gold medal, to be awarded by the Russian Academy of Sciences for distin- guished works in physics and astrophysics. • Name a street in Moscow after Ya. B. Zel- dovich. • Set the commemorative plaques in mem- ory of the Academician Ya. B. Zeldovich on the buildings of: – the M. V. Keldysh Institute of Ap- plied Mathematics of RAS – the Space Research Institute of RAS – the N. N. Semyonov Institute of Chemical Physics of RAS

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